Skip to main content

On the application of the Arlequin method to the coupling of particle and continuum models

Computational Mechanics volume 42, pages 511–530 (2008)Cite this article

Abstract

In this work, we propose to extend the Arlequin framework to couple particle and continuum models. Three different coupling strategies are investigated based on the L 2 norm, H 1 seminorm, and H 1 norm. The mathematical properties of the method are studied for a one-dimensional model of harmonic springs, with varying coefficients, coupled with a linear elastic bar, whose modulus is determined by simple homogenization. It is shown that the method is well-posed for the H 1 seminorm and H 1 norm coupling terms, for both the continuous and discrete formulations. In the case of L 2 coupling, it cannot be shown that the Babuška–Brezzi condition holds for the continuous formulation. Numerical examples are presented for the model problem that illustrate the approximation properties of the different coupling terms and the effect of mesh size.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

39,95 €

Price includes VAT (Finland)

References

  1. Babuška I (1973). The finite element method with Lagrangian multipliers. Numer Math 20: 179–192

    Article  Google Scholar 

  2. Belytschko T and Xiao SP (2003). Coupling methods for continuum model with molecular model. Int J Multiscale Comput Eng 1(1): 115–126

    Article  Google Scholar 

  3. Ben Dhia H (1998). Multiscale mechanical problems: the Arlequin method. C R Acad Sci Paris Sér IIB 326(12): 899–904

    MATH  Google Scholar 

  4. Ben Dhia H (2006). Global local approaches: the Arlequin framework. Eur J Comput Mech 15(1–3): 67–80

    Google Scholar 

  5. Ben Dhia H and Rateau G (2001). Mathematical analysis of the mixed Arlequin method. C R Acad Sci Paris Sér I 332: 649–654

    MATH  MathSciNet  Google Scholar 

  6. Ben Dhia H and Rateau G (2002). Application of the Arlequin method to some structures with defects. Rev Eur Eléments Finis 332: 649–654

    MathSciNet  Google Scholar 

  7. Ben Dhia H and Rateau G (2005). The Arlequin method as a flexible engineering design tool. Int J Numer Meth Eng 62(11): 1442–1462

    Article  MATH  Google Scholar 

  8. Brezzi F (1974). On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers. RAIRO Anal Numér 8(R-2): 129–151

    MathSciNet  Google Scholar 

  9. Broughton JQ, Abraham FF, Bernstein N and Kaxiras E (1999). Concurrent coupling of length scales: methodology and application. Phys Rev B 60(4): 2391–2403

    Article  Google Scholar 

  10. Ern A and Guermond JL (2004). Theory and practice of finite elements. Springer, New York

    MATH  Google Scholar 

  11. Fish J (2006). Bridging the scales in nano engineering and science. J Nanoparticle Res 8(6): 577–594

    Article  Google Scholar 

  12. Guidault P and Belytschko T (2007). On the L 2 and the H 1 couplings for an overlapping domain decomposition method using Lagrange multipliers. Int J Numer Meth Eng 70(3): 322–350

    Article  MathSciNet  Google Scholar 

  13. Liu WK, Karpov EG, Zhang S and Park HS (2004). An introduction to computational nanomechanics and materials. Comput Methods Appl Mech Eng 193: 1529–1578

    Article  MATH  MathSciNet  Google Scholar 

  14. Miller RE and Tadmor EB (2002). The quasicontinuum method: overview, applications and current directions. J Comput Aided Des 9: 203–239

    Article  Google Scholar 

  15. Oden JT, Prudhomme S, Romkes A and Bauman P (2006). Multi-scale modeling of physical phenomena: adaptive control of models. SIAM J Sci Comput 28(6): 2359–2389

    Article  MATH  MathSciNet  Google Scholar 

  16. Prudhomme S, Bauman PT and Oden JT (2006). Error control for molecular statics problems. Int J Multiscale Comput Eng 4(5-6): 647–662

    Article  Google Scholar 

  17. Prudhomme S, Ben Dhia H, Bauman PT, Elkhodja N, Oden JT (2008) Computational analysis of modeling error for the coupling of particle and continuum models by the arlequin method. Computer Methods in Applied Mechanics and Engineering (To appear)

  18. Wagner GJ and Liu WK (2003). Coupling of atomistic and continuum simulations using a bridging scale decomposition. J Comput Phys 190: 249–274

    Article  MATH  Google Scholar 

  19. Xiao SP and Belytschko T (2004). A bridging domain method for coupling continua with molecular dynamics. Comput Methods Appl Mech Eng 193: 1645–1669

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, TX, USA

    Paul T. Bauman, J. Tinsley Oden & Serge Prudhomme

  2. Laboratoire de Mécanique des Sols, Structures et Matériaux, Ecole Centrale de Paris, Châtenay-Malabry, France

    Hachmi Ben Dhia & Nadia Elkhodja

Authors
  1. Paul T. Bauman
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hachmi Ben Dhia
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Nadia Elkhodja
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. J. Tinsley Oden
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Serge Prudhomme
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Paul T. Bauman.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Bauman, P.T., Dhia, H.B., Elkhodja, N. et al. On the application of the Arlequin method to the coupling of particle and continuum models. Comput Mech 42, 511–530 (2008). https://doi.org/10.1007/s00466-008-0291-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-008-0291-1

Keywords

  • Multiscale modeling
  • Domain decomposition
  • Lagrange multipliers
  • Numerical methods
  • Atomistic-continuum coupling